Konstantinos A. Blekos
An algorithm is presented for unranking permutations in transposition order: Given a seed s\in N, the algorithm produces a permutation P(s) that differs from the permutation P(s+1) by the transposition of two elements.
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Ranking and Unranking Restricted Permutations
本文研究错排与 ménage 排列的高效反排序，将其归约为给定前缀的受限排列计数，并利用车理论求解，从而支持按字典序提取第 k 个排列。
We give some results about a bijection associating each permutation with a subexcedant function. This function is related to a particular decomposition of the permutation as a product of transpositions and therefore it has been called transposition array in the literature. In particular, we identify anti-exceedance positions of the permutation through its transposition array and we give an expression of the bijection in terms of the cycle structure of the permutation. We give a characterization of a family of permutations having a non-decreasing transposition array and study length 3 pattern a…
Multiset permutation generation by transpositions
本文提出一种仅用换位且优先相邻换位的多重集排列生成算法，并引入强齐次换位条件以限制非相邻换位，同时保持较小的存储开销。
We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored.
We compute the number of ways a given permutation can be written as a product of exactly $k$ transpositions. We express this number as a linear combination of explicit geometric sequences, with coefficients which can be computed in many particular cases. Along the way we prove several symmetry properties for matrices associated with bipartite graphs, as well as some general (likely known) properties of Young diagrams. The methods involve linear algebra, enumeration of border strip tableau, and a differential operator on symmetric polynomials.
Upper Embeddability of Graphs and Products of Transpositions Associated with Edges
本文研究图的边序如何决定由边关联对换乘积得到的顶点置换，利用图嵌入刻画了存在边序使该置换为完全循环置换的图，并给出 Eden 关于恒等置换问题的一个反例。
We consider the problem of factoring permutations as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances. In particular, we investigate the minimum number, $δ$, such that every permutation can be factored into no more than $δ$ special transpositions. This study is related to sorting algorithms, Cayley graphs, and genomics.
We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how man…
The reconstruction problem for permutations on $n$ elements from their erroneous patterns which are distorted by transpositions is presented in this paper. It is shown that for any $n \geq 3$ an unknown permutation is uniquely reconstructible from 4 distinct permutations at transposition distance at most one from the unknown permutation. The {\it transposition distance} between two permutations is defined as the least number of transpositions needed to transform one into the other. The proposed approach is based on the investigation of structural properties of a corresponding Cayley graph. In …
We consider the problem of upper bounding the number of circular transpositions needed to sort a permutation. It is well known that any permutation can be sorted using at most $n(n-1)/2$ adjacent transpositions. We show that, if we allow all adjacent transpositions, as well as the transposition that interchanges the element in position 1 with the element in the last position, then the number of transpositions needed is at most $n^2/4$. This answers an open question posed by Feng, Chitturi and Sudborough (2010).
Conjugating full cycles by adjacent transpositions: diameter and sorting time
研究通过相邻交换 conjugate 循环排列的步骤数上限和 Schreier 图直径，探讨排序时间与直径的关系。
In this paper, we introduce plane permutations, i.e. pairs $\mathfrak{p}=(s,π)$ where $s$ is an $n$-cycle and $π$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $π$) and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plan…
A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists of block transpositions. It should be noted that "the block transposition distance of a permutation" is the distance of the permutation from the identity permutation in the Cayley graph $\Cay$, and "sorting a permutation by block transpositions" is equivalent to finding shortest paths in $\Cay$. The original results in our thesis concern the lower and upper b…
A Genome rearrangement problem studies large-scale mutations on a set of DNAs in living organisms. Various rearrangements like reversals, transpositions, translocations, fissions, fusions, and combinations and different variations have been studied extensively by computational biologists and computer scientists over the past four decades. From a mathematical point of view, a genome is represented by a permutation. The genome rearrangement problem is interpreted as a problem that transforms one permutation into another in a minimum number of moves under certain constraints depending on the chos…
In genome rearrangements, the mutational event transposition swaps two adjacent blocks of genes in one chromosome. The Transposition Distance Problem (TDP) aims to find the minimum number of transpositions required to transform one chromosome into another, both represented as permutations. The TDP can be reduced to the problem of Sorting by Transpositions (SBT). SBT is $\mathcal{NP}$-hard and the best approximation algorithm with a $1.375$ ratio was proposed by Elias and Hartman. Their algorithm employs simplification, a technique used to transform an input permutation $π$ into a simple permut…
Signed Graphs and Signed Cycles of Hyperoctahedral Groups
本文将Dénes关于图边序与全循环置换的经典结果推广到有符号图和超八面体群，给出了有符号图存在偶或奇全循环边序的充要条件，并统计了奇全循环置换的最少对换表示数目。
We show that any smooth permutation $σ\in S_n$ is characterized by the set ${\mathbf{C}}(σ)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{\leqσ}$, and that $σ$ is the product (in a certain order) of the transpositions in ${\mathbf{C}}(σ)$. We also characterize the image of the map $σ\mapsto{\mathbf{C}}(σ)$. As an application, we show that $σ$ is smooth if and only if the intersection of $(S_n)_{\leqσ}$ with every conjugate of a parabolic subgroup of $S_n$ admits a maximum. This also gives another approach for enumerating smooth permutations and subclasses thereof. Finally, w…
In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $ω$, for which $ω(12\cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It turns out th…
Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every permutation $π\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(π)$ is associated with each $π\in{\rm{Sym}_n}$ taking the length of the shortest such word, and the $S$-diameter $d(S)$ is the maximum value of $d(π)$ with $π$ ranging over ${\rm{Sym_n}}$. The distance $d(π,ν)$ of two permutations $π,ν$ defined by $d(ν^{-1}\circπ)$ satisfies the axioms of a metric space. In this paper we consider the case where $S$ consists of all block transpositions of ${\rm{Sym_n}}$ and call $d(π)$ the block tra…
Given integers $k\geq 2$ and $a_1,\ldots,a_k\geq 1$, let $\boldsymbol{a}:=(a_1,\ldots,a_k)$ and $n:=a_1+\cdots+a_k$. An $\boldsymbol{a}$-multiset permutation is a string of length $n$ that contains exactly $a_i$ symbols $i$ for each $i=1,\ldots,k$. In this work we consider the problem of exhaustively generating all $\boldsymbol{a}$-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations …
Twisted Bracelets for Sorting by Transpositions: the Transposition Diameter of $S_{16}$
本文通过扭转手镯与扩展环面等价类对应，结合不动点收缩和结构归约及穷举验证，证明了 $S_{16}$ 的转位直径为 9，解决了 25 年未决问题。
Circular sorting in the alternating group
该研究探讨交错群中用3-循环排序偶置换的最大步数，针对不同模4余值的$n$确定或分析了排序次数，证明较大值会无限次出现。
We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations of length n which can be obtained by a finite number of block sorting operations (e.g., reversals, block transpositions, cut-and-paste moves).
We show that the random transposition walk on the symmetric group $S_n$ has cutoff in separation distance at $\frac{1}{2}n \log n$, by constructing a strong stationary time. The construction involves working with cycle types of permutations and some partition combinatorics.
An episode of Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, "Can the switching be undone so as to restore all minds to their original bodies?" Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two "outsiders." We refine Keeler's result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences o…
In the context of the genome rearrangement problem, we analyze two well known models, namely the block transposition and the prefix block transposition models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call generating permutations and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.
Circular sorting, strong complete mappings and wreath product constructions
研究在圆上通过交换标签排序标记点的最少交换次数，证明当n不是素数时最多需要n-3次交换，并构造需要n-2次交换的循环排列（n为素数），反驳了第二个猜想。
Short reachability networks
本文推广了置换网络，提出 $t$-可达性网络，解决了 $t=2$ 时的最短长度问题，并为固定 $t \geq 3$ 提供了随机化构造及星型对换限制的研究。
Balanced Gray Codes for Permutations and Rainbow Cycles for Associahedra
本研究解决了排列的平衡转置格雷码构造问题，并推广了彩虹环概念，为结合多面体和排列多面体提供了新的彩虹环构造，同时补充了偶数情况下的平衡格雷码。
A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The dimension (i.e. size) of each permutation grows by coupled Chinese Restaurant Processes, while in time each permutation evolves according to the random transposition chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in…
We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation. For every reduced word of the reverse permutation there is exactly one valid way to assign probabilities to the transpositions. It is an open problem to determine the minimum length of such a sequence when the simplicity condition is dropped.
Let $Γ$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $Γ$ is bounded as $$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$ where the maximization is over all permutations $π$, $c(π)$ denotes the number of cycles in $π$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upp…
Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences leading to the binary strings avoiding $1^k$. We prove a bijection between the set %$F_n^{(k)}$ of strings of length $n$ and the set of permutations of $S_{n+1}(321,312,23\ldots(k+1)1)$. Finally, basing on a known Gray code for those strings, we define a Gray code for $S_{n+1}(321,312,23\ldots(k+1)1)$, where two consecutive permutations differ by an adjacent tran…
Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $β$ in terms of blocks in cycles in the decomposition of $β$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute …
A factorization of a permutation into transpositions is called "primitive" if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on $n$ elements. Consider this walk in continuous time starting at the identity and let $D_t$ be the minimum number of transpositions needed to go back to the identity from the location at time $t$. $D_t$ undergoes a phase transition: the distance $D_{cn/2} \sim u(c)n$, where $u$ is an explicit function satisfying $u(c)=c/2$ for $c \le 1$ and $u(c)1$…
Random permutations from $q$-Demazure products
研究$q$-变形的Demazure乘积模型，通过删除转位并应用$q$-Demazure乘积得到排列，证明其极限为确定性permuton，解决猜想并明确极限permuton。
Improving the Gilbert-Varshamov bound for permutation Codes in the Cayley metric and Kendall $τ$-Metric
本文利用图论方法，将凯莱度量和Kendall τ-度量下置换码的Gilbert-Varshamov界分别渐近改进了一个log(n)因子。
List and total colorings of multiset permutation graphs
研究证明多重排列图ST_k^ℓ在特定条件下可被(ℓ-1)-可列表着色，并允许全着色，同时探讨了高效支配集的概念及其在排列图中的应用。
We prove that the polynomial invariants of a permutation group are Cohen-Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The "if" direction of the argument uses Stanley-Reisner theory and a recent result of Christian Lange in orbifold theory. The "only-if" direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstru…
In this paper we present a simple framework to study various distance problems of permutations, including the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. These problems are very important in the study of the evolution of genomes. We give a general formulation for lower bounds of the transposition and block-interchange distance from which the existing lower bounds obtained by Bafna and Pevzner, and Christie can be easily derived. As to the reversal distance of signed permutations, we translate it into a block-interchange …
Comparability of random permutations in the strong Bruhat order
研究随机排列在强布雷特顺序中的可比性，证明其概率下降速度比多项式更快，达到亚指数级别。
In this paper, we construct a listing for the vertices of the augmentation graph of given size, and as a consequence, we obtain a Hamiltonian cycle for the square of the augmentation graph of given size. As applications, we have a Gray code for the $132$-$312$ avoiding permutations of given length such that two successive permutations differ by at most $2$ adjacent transpositions. Also we obtain Gray codes of strong distance $2$ for the $001$ avoiding ascent sequences and the $010$ avoiding ascent sequences of given length.
Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly (i.e., the acceleration drops from 0 to -\infty at this time as n tends to \infty). On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is both considerably simpler and more general than in a previous result of Oded Schramm (2005) for random transpositions. It turns out t…
In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the k'th card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random shuffle was raised by Aldous and Diaconis in 1986. Recently, Mironov used this shuffle as a model for the cryptographic system known as ``RC4'' and proved an upper bound of O(n log n) for the mixing time. We prove a matching lower bound, thus establishing that the mixing time is indeed of order $Θ(n \log n)$. We also prove an upper bound of O(n log n) for the mixi…
Cuntz algebra automorphisms: transpositions
本文研究 Cuntz 代数的置换自同构，通过刻画 $S([n]^3)$ 中换位的稳定性，构造出任意 $n>1$ 时含 6 个自由度的新自同构族。
Let P be a permutation expressed as a product of nontrivial disjoint cycles. When writing P as a product of distinct transpositions none equal to a factor of P, what is the smallest number of transpositions that can be used? We answer this question and give applications to mind-switching problems that have arisen in connection with the popular sci-fi television series Futurama and Stargate SG-1.
Turning cycle restrictions into mesh patterns via Foata's fundamental transformation
本文通过Foata基本变换，将排列中相邻q-循环的数量映射为两种网格模式的出现次数之和，并以此解决了第一作者先前论文中的猜想3.14。
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ an…
Random sampling of permutations through quantum circuits
本文受 Steinhaus-Johnson-Trotter 算法启发，提出经典及量子电路模型下的排列随机采样算法，并应用于双样本随机化检验，同时提出嵌套 corona 乘积图生成模型以支持特定排列集合的采样。
In this paper, we study permutations $π\in S_n$ with exactly $m$ transpositions. In particular, we are interested in the expected value of $π(1)$ when such permutations are chosen uniformly at random. When $n$ is even, this expected value is approximated closely by $(n+1)/2$, with an error term that is related to the number isometries of the $(n/2-m)$-dimensional hypercube that move every face. Furthermore, when $k \mid n$, this construction generalizes to allow us to compute the expected value of $π(1)$ for permutations with exactly $m$ $k$-cycles. In this case, the expected value has an erro…
In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris--Kleitman sense). Thus, for example, for a (uniformly) random permutation $π$, the event that no point is displaced by more than a fixed distance $d$ and the event that $π$ is the product of at most $k$ adjacent transpositions are positively correla…
A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley networks. The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper bound given in the oft-cited paper of Akers and Krishnamurthy (1989). In this work, we first assess the performance of their upper bound. We show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Evaluating their upper bound takes time $Ω(n!)$. In this paper,…
Homological algebra of Nakayama algebras and 321-avoiding permutations
本文通过线性Nakayama代数与Dyck路径及321-避免排列的自然对应，给出了不动点统计量的同调解释，并证明了Jacobson根的自我扩张空间同构于$K^{\mathfrak{s}(\pi)}$。
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